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### Open Access Journals

DCDS

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the $2$-sphere, we show that for each $n$, $f$ has at least $2^{n}$ periodic points of period $n$.

DCDS-B

In this paper we consider the family of circle maps
$f_{k,\alpha,\epsilon}:\mathbb{S}^1\rightarrow \mathbb{S}^1$ which when
written mod 1 are of the form
$f_{k,\alpha,\epsilon}: x \mapsto k x + \alpha + \epsilon \sin(2\pi
x)$, where the parameter $\alpha$ ranges in $\mathbb{S}^1$
and $k\geq 2.$ We prove that for small $\epsilon$ the average over $\alpha$ of the
entropy of $f_{k,\alpha,\epsilon}$ with respect to the natural absolutely
continuous measure is smaller than $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx,$ while
the maximum with respect to $\alpha$ is larger. In the case of the average the
difference is of order of $\epsilon^{2k+2}.$ This result is in contrast to families
of expanding Blaschke products depending on rotations where the averages are
equal and for which the inequality for averages goes in the other direction when
the expanding property does not hold, see [4]. A striking fact for both
results is that the maximum of the entropies is greater than or equal to
$\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx$. These results should also be compared with
[3], where similar questions are considered for a family of
diffeomorphisms of the two sphere.

DCDS

none

DCDS

In [PS] it is conjectured that among the volume preserving $C^2$ diffeomorphisms of a closed manifold which have some hyperbolicity, the ergodic ones contain an open and dense set. In this paper we prove an analogous statement for skew products of Anosov diffeomorphisms of tori and circle rotations. Thus this paper may be seen as an example of the phenomenon conjectured in [PS]. The corresponding theorem for skew products of Anosov diffeomorphisms and translations of arbitrary compact groups is an interesting open problem.

JMD

We investigate transverse Hölder regularity of some canonical leaf
conjugacies in normally hyperbolic dynamical systems and transverse
Hölder regularity of some invariant foliations. Our results
validate claims made elsewhere in the literature.

DCDS

In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself),
[M]. Many generalizations of this Mautner phenomenon exist in
representation theory, [St1]. Here, we establish a new
generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.

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